Therefore we have bisected the given angle BAC by the straight line AF. Q.E.F.

So much for bisecting any angle. One of the "problems of antiquity" was to trisect any angle. Scholars, both professional and amateur, learned much about geometry as they struggled with the task for more than 2000 years, and some still do. It was not until the 19th century that it was proved that the trisection of any angle, using straightedge and compass, is impossible.

In the proof of Proposition 9, note that the side EF is called the base, as is the side DF. Why? Because Proposition 8 -- S.S.S. -- which allows us to conclude that those angles are equal, enunciates that the bases are also equal. By quoting the exact words of a previous proposition in that way, in a virtually legalistic manner, we make the sequence of reasoning perfectly clear.

To bisect a straight line

To bisect the straight line AB, place the point of the compass at A, and with radius AB draw an arc. Now place the point of the compass at B, and with the same radius draw an arc. Upon connecting their points of intersection C and E, the line CE cuts AB into two equal parts at D.